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Alterations of algebraic varieties (after A. J. de Jong). (Altérations de variétés algébriques (d’après A. J. de Jong).) (French) Zbl 0924.14007

Séminaire Bourbaki. Volume 1995/96. Exposés 805–819. Paris: Société Mathématique de France, Astérisque. 241, 273-311, Exp. No. 815 (1997).
This is a report on important recent results on resolution of singularities of algebraic varieties and the theory of semi-stable reduction, found in the paper “Smoothness, semi-stability and alterations”, by A. J. de Jong [Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996; Zbl 0916.14005)]. The key notion is “alteration”: An alteration of an integral noetherian scheme \(X\) is a proper, surjective morphism \(\varphi:X'\to X\) (with \(X'\) integral, noetherian) such that, for a suitable open dense set \(U\subseteq X\), the induced morphism \(\varphi_U:\varphi^{-1}(U)\to U\) is finite. For instance, and working (to simplify) with projective varieties over a field \(k\), the author proves: Given an integral variety \(X\) and a closed subvariety \(Z\subset X\), there is an alteration \(\pi:X'\to X\) with \(X'\) regular, such that \(\pi^{-1} (Z)_{\text{red}}\) is a strict normal crossings divisor (i.e., the irreducible components are smooth and meet transversally). Note that there are no restrictions on the base field \(k\).
This is the method of proof: Obtain an alteration \(f:X_1\to X\) such that there is flat morphism \(g:X_1\to T\), with \(T\) regular, \(\dim(T)= \dim(X_1)-1\), such that the general fiber of \(g\) is regular and any fiber is a curve with, at worst, ordinary double points as singularities, moreover the set of points of \(T\) where the fiber of \(g\) is singular is contained in a strict normal crossings divisor. Then, to desingularize such a \(X_1\) by means of monoidal transformations is easy. An alteration as above is obtained by using some classical projective techniques, the theory of moduli for pointed semi-stable curves and an induction hypothesis (applied to \(T\), whose dimension is one less than that of \(X)\).
A. J. de Jong also proved (loc. cit.) a weak version of the general semi-stable reduction theorem where (essentially) one allows to substitute one of the relevant varieties involved by an alteration thereof. These results (of course, precisely stated) are discussed in this report. There is an essentially complete proof of the desingularization theorem and good sketch of the one for the reduction problem. The report is an excellent introduction to these topics.
But there is also a very useful section (the last one) on applications. P. Berthelot discusses three:
(1) O. Gabber’s affirmative solution to Serre’s problem on multiplicity of intersection for two modules over a local noetherian ring \(A\): “\(\chi_A(M,N)\geq 0\)” (no restrictions on the ring);
(2) a proof (due to Berthelot) showing that the Monsky-Washnitzer cohomology groups \(H^n_{MW} (X/K)\) (where \(X\) is a smooth affine scheme over a field \(k\) with \(\text{ch} (k)>0\), \(K\) the fractions of a Cohen ring of \(k)\) are finite dimensional vector spaces over \(K\);
(3) some recent work of Deligne on monodromy actions on étale cohomology groups (specially, an “independence of \(l\)” theorem).
All these results use the desingularization theorem of de Jong.
For the entire collection see [Zbl 0866.00026].

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14F30 \(p\)-adic cohomology, crystalline cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies

Citations:

Zbl 0916.14005
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